week 4

Question 1

Define y_t for stock price using GARCH

Answer 1

Question 2

Properties of continuously compounded returns

Answer 2

Symmetric dist about mean for y_t (drift)

Little autocorrelation in y_t (hence ARMA inappropriate)

Strong autocorrelation in y_t²

Variable volatility: local variance of process changes substantially across time

Question 3

Formula for kurtosis

Answer 3

Question 4

Calculate MGF for Gaussian RV

Answer 4

Question 5

ARCH model

Answer 5

Crucially (G)ARCH models are defined by BOTH equations (which makes conversion to ARMA easier)

Question 6

GARCH(p, q)

Answer 6

Question 7

When to use EGARCH and why

Answer 7

When you suspect neg shocks may have diff effect size to pos shocks

When you don’t want to constrain params to no negativity (as log allows neg)

For modelling volatility

If you suspect volatility shocks have a large lasting impact, due to long memory of model

Question 8

Express GARCH(1,1) as ARMA(1,1)

Answer 8

vt is a martingale difference (E = 0)

Question 9

Estimation with GARCH and why

Answer 9

Question 10

Significance of Kurtosis

Answer 10

In ARCH or GARCH models, if standardised residuals have high kurtosis it suggest that model may not be capturing dynamics of time series

Question 11

Generally when to use ARCH and GARCH

Answer 11

When time series exhibits volatility clustering, periods of high volatility followed by low volatility

Question 12

Finding γ or ρ for (G)ARCH models

Answer 12

Put into ARMA form as it is easier (?)

Question 13

EGARCH in ARMA form

Answer 13

g(x) = ωx + λ(|x| - E_φ|x|)